Optimal. Leaf size=32 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x+b x^4}}\right )}{3 \sqrt {b}} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2029, 206} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x+b x^4}}\right )}{3 \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2029
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a x+b x^4}} \, dx &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a x+b x^4}}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x+b x^4}}\right )}{3 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 1.91 \[ \frac {2 \sqrt {x} \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a+b x^3}}\right )}{3 \sqrt {b} \sqrt {x \left (a+b x^3\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 94, normalized size = 2.94 \[ \left [\frac {\log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - a^{2} - 4 \, {\left (2 \, b x^{4} + a x\right )} \sqrt {b x^{4} + a x} \sqrt {b}\right )}{6 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {b x^{4} + a x} \sqrt {-b} x}{2 \, b x^{3} + a}\right )}{3 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 23, normalized size = 0.72 \[ -\frac {2 \, \arctan \left (\frac {\sqrt {b + \frac {a}{x^{3}}}}{\sqrt {-b}}\right )}{3 \, \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 979, normalized size = 30.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {b x^{4} + a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{\sqrt {b\,x^4+a\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {x \left (a + b x^{3}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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